Polynomials for Best Approximation over Semi-Infinite and Infinite Intervals

Abstract

It is well hlown that Chebyshev polynomials play a useful key role in giting close polynomial approximations to functions defined over finite intervalsBy employing Chebyshev polynomials, we can approximate a polynomial of very high degree (which'is practically the same as any continuous function) by another polynomial of rnuch lower degree, with great accuracy. This approximation is a consequence of the property that, for any finite interval [a,b], for fixed degree n, the Chebyshes polynomial (normalized to [a,b] ) has a maximum deviation from zero which is less than that of any other polynomial of degree n, and having the same leading coefficient of xn. But Chebyshev polynomials do not serve as well in approximations over infinite intervals, where an approximating polynomial will usually require a damping or weight factor. The simplest analogue of Chebyshev polynomials over infinite intervals would be polynomials harring th property that, for fixed degree n, when multiplied by e x or e x for [0,] or [_a),a)] respectively, they have a maximum deviation from zero which is less than